Christophe LETELLIER
16/04/2009

Otto E. Rössler & Peter J. Ortoleva

**What torn unimodal chaos is ?**

It is known that for producing a chaotic behavior, sensitivity to initial conditions is combined to some recurrence properties. These two specific characteristics result from two mechanisms : stretching and squeezing. This can be produced by a folding or a tearing. Typically, an attractor involving a folding is produced by the Rössler system and one involving a tearing is the Lorenz system. These two mechanisms were investigated in [1]. *Torn Unimodal chaos* corresponds to an attractor with a tearing mechanism that is characterized by a cusp --- or a Lorenz --- map. The Lorenz system is a good example but it has a rotation symmetry. The purpose here is to have an attractor with a tearing mechanism without any symmetry.

**The system**

To the best of our knowledge, the first set of equations that was identified to produce a chaotic attractor bounded by a genus-1 torus and possessing a Lorenz map was proposed by Otto Rössler and Peter Ortoleva from Indiana University [2] as an isothermal abstract reaction system. The systems reads :

This abstract chemical reaction produces a torn unimodal chaotic
attractor as shown in Fig. 1. Parameter values are *a*=33, *b*=150, *c*=1, *d*=3.5, *e*=4815, *f*=410, *g*=0.59, *h*=4, *j*=2.5, *k*=2.5, *l*=5.29, *m*=750, *K*_{1}=0.01 and
*K*_{2}=0.01. A first-return map to a Poincaré
section (Fig. 2) has the shape of the Lorenz map as expected.
The *l*-value is slightly modified to obtain a Lorenz map without a gap between the two monotonic branches as originally published [2].

**Fig. 1 : Chaotic attractor with a tearing mechanism**

**Fig. 2 : Lorenz map with its characteristic cusp**

[1] **G. Byrne, R. Gilmore & C. Letellier**,
Distinguishing between folding and tearing mechanisms in strange attractors, *Physical Review E*, **70**, 056214, 2004.

[2] **O. E. Rössler & P. J. Ortoleva**, Strange attractors in 3-variable reaction systems, *Lecture Notes in Biomathematics*, **21**, 67-73, 1978.